diff --git a/slides/cours_21.md b/slides/cours_21.md index 300abae67cf6798cedba2fc1a2fffccb539c087f..761f0e655b21602eee433f482f5e036209a09c47 100644 --- a/slides/cours_21.md +++ b/slides/cours_21.md @@ -273,6 +273,8 @@ rien moyenne(arbre) { # La compression avec pertes +\footnotesize + ## Pseudo-code (5min, matrix) . . . diff --git a/slides/cours_22.md b/slides/cours_22.md new file mode 100644 index 0000000000000000000000000000000000000000..47d8b8c3512a05f0e0563347a92117f1200990ac --- /dev/null +++ b/slides/cours_22.md @@ -0,0 +1,366 @@ +--- +title: "B-Arbres" +date: "2022-04-13" +patat: + eval: + tai: + command: fish + fragment: false + replace: true + ccc: + command: fish + fragment: false + replace: true + images: + backend: auto +--- + +# Le cours précédent + +* Questions: + + + +# Les B-arbres + +## Problématique + +* Grands jeux de données (en 1970). +* Stockage dans un arbre, mais l'arbre tiens pas en mémoire. +* Regrouper les sous-arbres en **pages** qui tiennent en mémoire. + +## Exemple + +* 100 noeuds par page et l'arbre comporte $10^6$ noeuds: + * Recherche B-arbre: $\log_{100}(10^6)=3$; + * Recherche ABR: $\log_2(10^6)=20$. +* Si on doit lire depuis le disque: $10\mathrm{ms}$ par recherche+lecture: + * $30\mathrm{ms}$ (lecture beaucoup plus rapide que recherche) vs $200\mathrm{ms}=0.2\mathrm{s}$. + +## Remarques + +* On sait pas ce que veut dire `B`: Bayer, Boeing, Balanced? +* Variante plus récente B+-arbres. + +# Les B-arbres + +## Illustration, arbre divisé en pages de 3 noeuds + + + +. . . + +## Utilisation + +* Bases de données (souvent très grandes donc sur le disque); +* Système de fichier. + +# Les B-arbres + +## Avantages + +* Arbres moins profonds; +* Diminue les opération de rééquilibrage; +* Complexité toujours en $\log(N)$; + +. . . + +## Définition: B-arbre d'ordre $n$ + +* Chaque page d'un arbre contient au plus $2\cdot n$ *clés*; +* Chaque page (excepté la racine) contient au moins $n$ clés; +* Chaque page qui contient $m$ clés contient soit: + * $0$ descendants; + * $m+1$ descendants. +* Toutes les pages terminales apparaissent au même niveau. + +# Les B-arbres + +## Est-ce un B-arbre? + + + +. . . + +## Oui! + +* Dans chaque noeud les clés sont **triées**. +* Chaque page contient au plus $n$ noeuds: check; +* Chaque noeud avec $m$ clés a $m+1$ descendants; +* Toutes les feuilles apparaissent au même niveau. + +# Les B-arbres + +## Exemple de recherche: trouver `32` + + + +. . . + +* Si `n` plus petit que la 1e clé ou plus grand que la dernière descendre. +* Sinon parcourir (par bissection ou séquentiellement) jusqu'à trouver ou descendre entre 2 éléments. + +# Les B-arbres + +## La recherche de la clé `C` algorithme + +0. En partant de la racine. +1. Si on est dans une feuille: + * Si la `C` est dans une page, retourner la page; + * Sinon c'est perdu. +2. Sinon: + * Tant que `C > page` passer à la page suivante + * Descendre + +# Les B-arbres + +## Disclaimer + +* Inspiration de <https://en.wikipedia.org/wiki/B-tree> + +## Exemples d'insertion: `1` + + + +. . . + +* L'arbre est vide, on insère juste dans la première page. + +# Les B-arbres + +## Exemples d'insertion: `2` + + + +. . . + +* La première page est pas pleine, on insère dans l'ordre (après 1). + +# Les B-arbres + +## Exemples d'insertion: `3` + +{width=50%} + +* Comment on insère (1min de réflexion avant de donner une réponse!)? + +# Les B-arbres + +## Exemples d'insertion: `3` + +{width=50%} + +. . . + +* La page est pleine, on crée deux enfants. +* On choisit, `2`, la médiane de `1, 2, 3` et il est inséré à la racine. +* `1` descend à gauche, `3` descend à droite. + +# Les B-arbres + +## Exemples d'insertion: `4` + +{width=50%} + +* Comment on insère (1min de réflexion avant de donner une réponse!)? + +# Les B-arbres + +## Exemples d'insertion: `4` + +{width=50%} + +. . . + +* On pourrait insérer à droite de `2`, mais... ça ferait 2 parents pour 2 enfants (mais `m` parents => `m+1` enfants ou `0`); +* On descend à droite (`4 > 2`); +* On insère à droite de `3`. + +# Les B-arbres + +## Exemples d'insertion: `5` + +{width=50%} + +* Comment on insère (1min de réflexion avant de donner une réponse!)? + +# Les B-arbres + +## Exemples d'insertion: `5` + + + +. . . + +* On descend à droite (on peut pas insérer à la racine comme pour `4`); +* On dépasse la capacité de l'enfant droite; +* `4`, médiane de `3, 4, 5`, remonte à la racine; +* On crée un nouveau noeud à droite de `4`; +* La règle `m => m+1` est ok. + +# Les B-arbres + +## Exemples d'insertion: `6` + +{width=50%} + +* Comment on insère (1min de réflexion avant de donner une réponse!)? + +# Les B-arbres + +## Exemples d'insertion: `6` + + + +. . . + +* `6 > 4` on descend à droite; +* `6 > 5` et on a à la place à droite, on insère. + +# Les B-arbres + +## Exemples d'insertion: `7` + +{width=50%} + +* Comment on insère (1min de réflexion avant de donner une réponse!)? + +# Les B-arbres + +## Exemples d'insertion: `7` + +{width=50%} + +. . . + +* `7 > 4` on descend à droite; +* `7 > 6` mais on a dépassé la capacité; +* `6` est la médiane de `5, 6, 7`, remonte à la racine; +* `5` reste à gauche, `7` à droite, mais `6` fait dépasser la capacité de la racine; +* `4` est la médiane de `2, 4, 6`, `4` remonte, `2` reste à gauche, `6` à droite. + +# Les B-arbres + +## L'algorithme d'insertion + +0. Rechercher la feuille (la page a aucun enfant) où insérer; +1. Si la page n'est pas pleine insérer dans l'ordre croissant. +2. Si la page est pleine, on sépare la page en son milieu : + 1. On trouve la médiane, `M`, de la page; + 2. On met les éléments `< M` dans la page de gauche de `M` et les `> M` dans la page de droite de `M`; + 3. `M` est insérée récursivement dans la page parent. + +# Les B-arbres + +## Exercice: insérer `22` dans l'arbre d'ordre 2 (3min matrix) + + + +. . . + + + + +# Les B-arbres + +## Exercice: insérer `5, 45, 50` dans l'arbre d'ordre 2 (3min matrix) + + + +. . . + + + +# Les B-arbres + +## Exercice: insérer `32, 55, 60` dans l'arbre d'ordre 2 (3min matrix) + + + +. . . + + + +# Les B-arbres + +## Exercice: insérer `41` dans l'arbre d'ordre 2 (3min matrix) + + + +. . . + + + +# Les B-arbres + +## Exercice + +* Insérer 20, 40, 10, 30, 15, 35, 7, 26, 18, 22, 5, 42, 13, 46, 27, 8, 32, 38, 24, 45, 25, 2, 14, 28, 32, 41, +* Dans un B-arbre d'ordre 2. + +# Les B-arbres + +## Structure de données + +* Chaque page a une contrainte de remplissage, par rapport à l'ordre de l'arbre; +* Un noeud (page) est composé d'un tableau de clés/pointeurs vers les enfants; + +``` +P_0 | K_1 | P_1 | K_2 | | P_i 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